en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem
2 corrections found
The Campbell-Kelly theorem shows that Condorcet methods are the most spoiler-resistant class of ranked voting systems: whenever it is possible for some ranked voting system to avoid a spoiler effect, a Condorcet method will do so.
This misstates the Campbell–Kelly result. Campbell and Kelly’s theorem is a strategy-proofness characterization of majority rule on a Condorcet domain, not a theorem about spoiler effects or the comparative spoiler-resistance of Condorcet methods.
Full reasoning
The cited claim assigns the name Campbell–Kelly theorem to a result about spoiler effects and says it proves Condorcet methods are the most spoiler-resistant ranked methods. But the published Campbell & Kelly result is about something else.
The paper "A strategy-proofness characterization of majority rule" states in its abstract that majority rule is the only non-dictatorial and strategy-proof social choice rule on a domain where a strong Condorcet winner exists. That is a characterization theorem about strategy-proofness and majority rule on a restricted domain. It is not a theorem saying Condorcet methods are the uniquely most spoiler-resistant class of ranked voting systems, and it does not prove the quoted spoiler-effect claim.
So the sentence is not just imprecise; it attributes a different subject matter and conclusion to Campbell and Kelly's theorem than the source actually has.
1 source
- A strategy-proofness characterization of majority rule
Abstract: 'Majority rule is the only non-dictatorial and strategy-proof social choice rule' on the domain of profiles for which a strong Condorcet winner exists.
While Arrow's theorem does not apply to graded systems, Gibbard's theorem still does: no voting game can be straightforward (i.e. have a single, clear, always-best strategy).
This overstates Gibbard’s theorem. Gibbard–Satterthwaite applies only under additional conditions such as universal domain and more than two possible outcomes; strategy-proof voting rules can exist in restricted domains or dictatorial/two-outcome cases.
Full reasoning
The quoted sentence states Gibbard's theorem as a blanket claim that no voting game can be straightforward. That is too broad.
A standard statement of the theorem is conditional: on the universal domain, any strategy-proof voting rule whose range contains more than two alternatives must be dictatorial. In other words, the impossibility is not "no voting game can be straightforward" full stop. The theorem explicitly leaves exceptions, including dictatorial rules and rules with only two possible outcomes. It also does not rule out strategy-proof, non-dictatorial rules on restricted domains.
The Karlsruhe Institute of Technology summary states exactly this: on the universal domain, strategy-proof voting rules with more than two alternatives are dictatorial, and on restricted domains such as single-peaked preferences, one can find strategy-proof and non-dictatorial rules. The Stanford Encyclopedia likewise states the theorem with several conditions: universal domain, non-dictatorship, range constraint, resoluteness, and strategy-proofness cannot all be satisfied together.
So the article’s sentence replaces a conditional impossibility theorem with an unconditional universal claim, which is incorrect.
2 sources
- Social Choice Theory (Stanford Encyclopedia of Philosophy)
"There exists no social choice rule satisfying universal domain, non-dictatorship, the range constraint, resoluteness, and strategy-proofness."
- KIT - ECON Research - Research Areas - Social Choice Theory - Strategy-Proofness
"The famous Gibbard-Satterthwaite theorem states that on the universal domain all strategy-proof voting rules whose range contains more than two alternatives are dictatorial. One way of escaping dictatorship is by considering restricted preference domains... There one can find strategy-proof and non-dictatorial voting rules."